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SUMMARY:Juergen Herzog (University of Duisberg-Essen\, Germany)
DTSTART:20210729T082500Z
DTEND:20210729T085500Z
DTSTAMP:20260404T095208Z
UID:CommutativeAlgebra/1
DESCRIPTION:Title: A short survey of numerical semigroup rings\nby Juer
gen Herzog (University of Duisberg-Essen\, Germany) as part of Symposium i
n honour of Dilip Patil\n\n\nAbstract\nIn this lecture\, I will give a sho
rt survey on numerical semigroups from a\nviewpoint of commutative algebra
. A numerical semigroup is a subsemigroup\n$S$ of the additive semigroup o
f non-negative integers. One may assume\nthat the greatest common divisor
of the elements of $s$ is one. Then\nthere is an integer $F(S) \\not\\in S
$\, such that all integers bigger than\n$F(S)$ belongs to $S$. This number
is called the Frobenius number of $S$.\nFor a fixed field\, $K$ one consi
ders the $K$-algebra $K[S]$ which is the\nsubalgebra of the polynomial rin
g $K[t]$ which is generated over $K$ by\nthe powers $t^s$ with $s\\in S$.
This algebra is finitely generated and\nits relation ideal $I(S)$ is a bi
nomial ideal. In general\, it is hard to\ncompute $I(S)$. I will recall w
hat is known about this ideal by my own\nwork but also by the work of Bres
insky\, Delorme\, Gimenez\, Sengupta and\nSrinivasan\, Patil\, and others.
The semigroup ring $K[S]$ is a\nCohen--Macaulay domain\, and by the theor
em of Kunz it is Gorenstein if\nand only if the semigroup $S$ is symmetric
. Barucci\, Dobbs\, and Fontana\nintroduced pseudo-symmetric numerical sem
igroups. This concept was\ngeneralized by Barucci and Fröberg\, who intro
duced almost symmetric\nnumerical semigroups. The corresponding semigroup
ring is called almost\nGorenstein. Onc can define almost Gorenstein rings
not only in dimension\n$1$. A full-fledged theory in this direction has b
een developed by Goto\,\nTakahashi and Taniguchi. By considering the trace
of the canonical ideal\nof a numerical semigroup ring one is led to defin
e nearly Gorenstein\nnumerical semigroups\, as has been done by Hibi\, Sta
mate\, and myself. I\nwill briefly discuss these generalizations of Gorens
teiness and address a\nfew open problems related to this.\n
LOCATION:https://stable.researchseminars.org/talk/CommutativeAlgebra/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jugal K Verma (IIT Bombay\, India)
DTSTART:20210729T090000Z
DTEND:20210729T093000Z
DTSTAMP:20260404T095208Z
UID:CommutativeAlgebra/2
DESCRIPTION:Title: On the vanishing of the Chern number of filtrations of i
deals\nby Jugal K Verma (IIT Bombay\, India) as part of Symposium in h
onour of Dilip Patil\n\n\nAbstract\nLet $I$ be an $\\mathfrak m$-primary i
deal of a Noetherian local ring $R$.\nLet $\\mathcal F$ be an $I$-good fil
tration of ideals. The second Hilbert\ncoefficient $e_1(\\mathcal F)$ of t
he Hilbert polynomial of $\\mathcal F$\nis called its Chern number. We dis
cuss how the vanishing of the Chern\nnumber characterizes Cohen--Macaulay
local rings\, regular local rings and\n$F$-rational local rings using the
$I$-adic filtration\, the filtrations\nof the integral closure of powers\,
and the filtration of the tight\nclosure of powers of a parameter ideal.
We provide a partial answer to a\nquestion of Craig Huneke about $F$-ratio
nal local rings. \n\nThis is joint work with Saipriya Dubey (IIT Bombay) a
nd Pham Hung\nQuy (FPT University\, Vietnam).\n
LOCATION:https://stable.researchseminars.org/talk/CommutativeAlgebra/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Indranath Sengupta (IIT Gandhinagar\, India)
DTSTART:20210729T093500Z
DTEND:20210729T100500Z
DTSTAMP:20260404T095208Z
UID:CommutativeAlgebra/3
DESCRIPTION:Title: Some results on numerical semigroup rings\nby Indran
ath Sengupta (IIT Gandhinagar\, India) as part of Symposium in honour of D
ilip Patil\n\n\nAbstract\nWe will discuss Professor Patil's contribution i
n the field of numerical\nsemigroups and my association with the subject t
hrough some old and\nrecent results.\n
LOCATION:https://stable.researchseminars.org/talk/CommutativeAlgebra/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rajendra V Gurjar (IIT Bombay\, India)
DTSTART:20210729T102000Z
DTEND:20210729T105000Z
DTSTAMP:20260404T095208Z
UID:CommutativeAlgebra/4
DESCRIPTION:Title: On $\\mathbb A^1$-fibrations of affine varieties\nby
Rajendra V Gurjar (IIT Bombay\, India) as part of Symposium in honour of
Dilip Patil\n\n\nAbstract\nWe will begin with the fundamental result of Fu
jita--Miyanishi--Sugie that\na smooth affine surface $V$ has log Kodaira d
imension $-\\infty$ if and\nonly if $V$ has an $\\mathbb{A}^1$-fibration o
ver a smooth curve.\nGeneralizations of this to singular affine surfaces a
nd higher\ndimensional affine varieties raise non-trivial questions. We wi
ll\ndescribe some results in these directions. Connection with\nlocally-ni
lpotent derivations will be mentioned. Use of topological\narguments for p
roving some of these results will be indicated.\n
LOCATION:https://stable.researchseminars.org/talk/CommutativeAlgebra/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Martin Kreuzer (University of Passau\, Germany)
DTSTART:20210729T105500Z
DTEND:20210729T112500Z
DTSTAMP:20260404T095208Z
UID:CommutativeAlgebra/5
DESCRIPTION:Title: Differential methods for 0-dimensional schemes\nby M
artin Kreuzer (University of Passau\, Germany) as part of Symposium in hon
our of Dilip Patil\n\n\nAbstract\nGiven a $0$-dimensional subscheme $X$ in
$\\mathbb{P}^n$\, the traditional\nway to study the geometry of $X$ is to
look at algebraic properties of\nits homogeneous coordinate ring $R = K[x
_0\,\\ldots\,x_n]/I_X$ and the\nstructure of the canonical module of $R$.\
n\nHere we introduce and exploit a novel approach: we look at the Kaehler\
ndifferential algebra $\\Omega_{R/K}$ which is the exterior algebra over\n
the Kaehler differential module $\\Omega^1_{R/K}$ of $X$. Based on a\ncare
ful examination of the embedding of R into its normal closure and the\ncor
responding embedding of $\\Omega^1_{R/K}$\, we provide new bounds for\nthe
regularity index of the Kaehler differential module and connect it\nto th
e geometry of $X$ in low embedding dimensions.\n
LOCATION:https://stable.researchseminars.org/talk/CommutativeAlgebra/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Leslie Roberts (Queen's University\, Canada)
DTSTART:20210729T113000Z
DTEND:20210729T120000Z
DTSTAMP:20260404T095208Z
UID:CommutativeAlgebra/6
DESCRIPTION:Title: Ideal generators of projective monomial curves in $\\mat
hbb{P}^3$\nby Leslie Roberts (Queen's University\, Canada) as part of
Symposium in honour of Dilip Patil\n\n\nAbstract\nI discuss ideal generato
rs of projective monomial curves of degree $d$ in\n$\\mathbb{P}^3$\, based
on the paper P. Li\, D.P. Patil and L. Roberts\,\nBases and ideal generat
ors for projective monomial curves\,\nCommunications in Algebra\, 40(1)\,
pages 173--191\, 2012\, which was my last\npaper with Dilip. I also discus
s more recent observations by Ping Li and\nmyself.\n
LOCATION:https://stable.researchseminars.org/talk/CommutativeAlgebra/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shreedevi Masuti (IIT Dharwad\, India)
DTSTART:20210730T043000Z
DTEND:20210730T050000Z
DTSTAMP:20260404T095208Z
UID:CommutativeAlgebra/7
DESCRIPTION:Title: The Waring rank of binary binomial forms\nby Shreede
vi Masuti (IIT Dharwad\, India) as part of Symposium in honour of Dilip Pa
til\n\n\nAbstract\nIt is well known that every form $F$ of degree $d$ over
a field can be\nexpressed as a linear combination of $d$th powers of line
ar forms. The\nleast number of summands required for such an expression of
$F$ is known\nas the Waring rank of $F$. Computing the Waring rank of a f
orm is a\nclassical problem in mathematics. In this talk\, we will discuss
the Waring\nrank of binary binomial forms. This is my joint work with L.
Brustenga.\n
LOCATION:https://stable.researchseminars.org/talk/CommutativeAlgebra/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Parnashree Ghosh (ISI\, Kolkata\, India)
DTSTART:20210730T050500Z
DTEND:20210730T053500Z
DTSTAMP:20260404T095208Z
UID:CommutativeAlgebra/8
DESCRIPTION:Title: Homogeneous locally nilpotent derivations of rank 2 and
3 on $k[X\, Y\, Z]$\nby Parnashree Ghosh (ISI\, Kolkata\, India) as pa
rt of Symposium in honour of Dilip Patil\n\n\nAbstract\nIn this talk we wi
ll discuss homogeneous locally nilpotent derivations\n(LND) on $k[X\, Y\,
Z]$ where $k$ is a field of characteristic $0$. For a\nhomogeneous locally
nilpotent derivation $D$ on the polynomial ring in\nthree variables we wi
ll see how the $\\deg_D$ values of the linear terms\nare related and see a
consequence on the rank $3$ homogeneous derivations\nof degree $\\leq 3$.
\n\nFurther we will discuss homogeneous locally nilpotent derivations of r
ank\n$2$ and give a characterization of the triangularizable derivations a
mong\nthose. We will also see the freeness property of a homogeneous\ntria
ngularizable LND on $k[X\, Y\, Z]$.\n
LOCATION:https://stable.researchseminars.org/talk/CommutativeAlgebra/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kriti Goel (IIT Gandhinagar\, India)
DTSTART:20210730T054500Z
DTEND:20210730T061500Z
DTSTAMP:20260404T095208Z
UID:CommutativeAlgebra/9
DESCRIPTION:Title: On Row-Factorization matrices and generic ideals\nby
Kriti Goel (IIT Gandhinagar\, India) as part of Symposium in honour of Di
lip Patil\n\n\nAbstract\nThe concept of Row-factorization (RF) matrices wa
s introduced by A.\nMoscariello to explore the properties of numerical sem
igroups. For\nnumerical semigroups $H$ minimally generated by an almost ar
ithmetic\nsequence\, we give a complete description of the RF-matrices ass
ociated\nwith their pseudo-Frobenius elements. We use the information from
\nRF-matrices to give a characterization of the generic nature of the\ndef
ining ideal of the semigroup. Further\, when $H$ is symmetric and has\nemb
edding dimension 4 or 5\, we prove that the defining ideal is minimally\ng
enerated by RF-relations.\n\nThis is joint work with Om Prakash Bhardwaj a
nd Indranath Sengupta.\n
LOCATION:https://stable.researchseminars.org/talk/CommutativeAlgebra/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Neena Gupta (ISI\, Kolkata\, India)
DTSTART:20210730T062000Z
DTEND:20210730T065000Z
DTSTAMP:20260404T095208Z
UID:CommutativeAlgebra/10
DESCRIPTION:Title: On $2$-stably isomorphic four dimensional affine domain
s\nby Neena Gupta (ISI\, Kolkata\, India) as part of Symposium in hono
ur of Dilip Patil\n\n\nAbstract\nA famous theorem of Abhyankar--Eakin--Hei
nzer proves that if $A$ is a one\ndimensional ring containing $\\mathbb{Q}
$ and $n \\ge 1$ be such that the\npolynomial ring in $n$-variables over $
A$ is isomorphic to the polynomial\nring in $n$ variables over $B$ for som
e ring $B$\, then $A \\cong B$. This\ndoes not hold in higher dimensional
rings in general. In this connection\nthe following question arises:\n\nIf
$A[X\,Y] \\cong B[X\,Y]$\, does this imply $A \\cong B$?\n\nIn this talk
we shall present four-dimensional seminormal\naffine domains over ${\\math
bb C}$ for which the above question does not\nhold. \n\nThis is joint work
with Professor T. Asanuma.\n
LOCATION:https://stable.researchseminars.org/talk/CommutativeAlgebra/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dilip Patil (IISc\, Bangalore)
DTSTART:20210729T121000Z
DTEND:20210729T122500Z
DTSTAMP:20260404T095208Z
UID:CommutativeAlgebra/11
DESCRIPTION:Title: Reminiscences\nby Dilip Patil (IISc\, Bangalore) as
part of Symposium in honour of Dilip Patil\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/CommutativeAlgebra/11/
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